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Journal of Statistical Physics

, Volume 1, Issue 1, pp 57–69 | Cite as

Differential entropy and tiling

  • Edward C. Posner
  • Eugene R. Rodemich
Articles

Abstract

This paper relates the differential entropy of a sufficiently nice probability density functionp on Euclideann-space to the problem of tilingn-space by the translates of a given compact symmetric convex setS with nonempty interior. The relationship occurs via the concept of the epsilon entropy ofn-space under the norm induced byS, with probability induced byp. An expression is obtained for this entropy asε approaches 0, which equals the differential entropy ofp, plusn times the logarithm of 2/ε, plus the logarithm of the reciprocal of the volume ofS, plus a constantC(S) depending only onS, plus a term approaching zero withε. The constantC(S) is called the entropic packing constant ofS; the main results of the paper concern this constant. It is shown thatC(S) is between 0 and 1; furthermore,C(S) is zero if and only if translates ofS tile all ofn-space.

Key words

Differential entropy Tiling Entropy Close packing Random coding Convex sets Epsilon entropy Information-theoretic geometry 

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References

  1. 1.
    E. C. Posner, E. R. Rodemich, and H. Rumsey, Jr., “Epsilon entropy of stochastic processes,”Ann. Math. Stat. 38:1000–1020 (1967).Google Scholar
  2. 2.
    E. C. Posner and E. R. Rodemich, “Epsilon entropy and data compression,” in preparation.Google Scholar
  3. 3.
    C. E. Shannon, A mathematical theory of communication,Bell System Tech. J. 27:379–423 (1948).Google Scholar
  4. 4.
    F. Hausdorff,Set Theory (trans. from German), Chelsea, New York (1957).Google Scholar
  5. 5.
    L. Fejes Tóth,Regular Figures, Pergamon Press, New York (1964).Google Scholar
  6. 6.
    C. A. Rogers,Packing and Covering, Cambridge University Press (1964).Google Scholar

Copyright information

© Plenum Publishing Corporation 1969

Authors and Affiliations

  • Edward C. Posner
    • 1
  • Eugene R. Rodemich
    • 1
  1. 1.Jet Propulsion LaboratoryCalifornia Institute of TechnologyPasadena

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